REVERSIBLE LOGIC SYNTHESIS BY QUANTUM ROTATION GATES

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1 Quntum Inormtion nd Computtion, ol, No N YYYY inton Press EESIBLE LOGIC SYNTHESIS BY QANTM OTATION GATES AFSHIN ABDOLLAHI, MEHDI SAEEDI, MASSOD PEDAM Deprtment o Eletril Enineerin, niversity o Soutern Cliorni Los Aneles, CA A rottion-sed syntesis rmewor or reversile loi is proposed We develop nonil representtion sed on inry deision dirms nd introdue opertors to mnipulte te developed representtion model Furtermore, reursive untionl ideomposition ppro is proposed to utomtilly syntesize iven untion Wile Boolen reversile loi is prtiulrly ddressed, our rmewor onstruts intermedite quntum sttes tt my e in superposition, ene we omine teniques rom reversile Boolen loi nd quntum omputtion Te proposed ppro results in polynomil te ount or multiple-ontrol Tooli tes witout nille were te previous ppro uses eponentil numer o tes We lso improve iruit dept or quntum rry-ripple dder y onstnt tor, nd iruit size or quntum multipleer rom On to On lo n Keywords: Quntum iruits, reversile iruits, rottion tes, inry deision dirms Communited y: to e illed Introdution Te ppel or reser on quntum inormtion proessin [] is due to tree mjor resons Worin wit inormtion enoded t te tomi sle su s ions nd even elementry prtiles su s potons is sientii dvne Diret mnipultion o quntum inormtion my rete new pilities su s ultr-preise mesurement [], telemetry, nd quntum litorpy [3], nd omputtionl simultion o quntum-menil penomen [4] 3 Some time-eponentil omputtionl tss wit non-quntum input nd output ve eiient quntum loritms [] Prtiulrly, most quntum iruits ieve quntum speed-up over onventionl loritms [5] However, useul pplitions remin limited eent dvnes in ult-tolernt quntum omputin derese per-te error rtes elow te tresold estimte [6] promisin lrer quntum omputin systems To e le to do eiient quntum omputtion, one needs to ve n eiient set o omputer-ided desin tools in ddition to te ility o worin wit vorle ompleity lss nd ontrollin quntum menil systems wit i idelity nd lon oerene times Tis is omprle wit te lssil domin were Turin mine, i lo speed nd no errors in switin were not dequte to desin st modern omputers Quntum iruit desin wit loritmi teniques nd CAD tools s een ollowed y severl reserers Te proposed metods eiter ddressed permuttion mtries [7] Current ddress: Knit Cpitl Ameris LLC, 545 Wsinton Blvd, Jersey City, NJ 0730 Correspondin utor: mseedi@usedu

2 eversile Loi Syntesis y Quntum ottion Gtes or unitry mtries, e, [8] Permuttion mtries nd reversile iruits re n importnt lss o omputtions tt sould e eiiently perormed or te purpose o eiient quntum omputtion Indeed, Boolen reversile iruits ve ttrted ttention s omponents in severl quntum loritms inludin Sor s quntum torin [9, 0] nd stilizer iruits [] In tis pper, nonil deision dirm-sed representtion is presented wit novel teniques or syntesis o iruits wit inry inputs Tis wor my e onsidered lon wit te wor done or te syntesis o reversile iruits [7] However, we wor wit rottionsed tes wi llow omputin Boolen untion y levin te Boolen domin [] Tereore, tis ppro my e viewed s step to eplore syntesis o reversile untions y tes oter tn enerlized Tooli nd Fredin tes We sow tt pplyin te proposed ppro improves iruit size or multiple-ontrol Tooli tes rom eponentil in [3] to polynomil, iruit dept or quntum rry-ripple dders y onstnt tor ompred to [4], nd 3 iruit size or quntum multipleers rom On to On lo n Te reminder o tis pper is ornized s ollows In Setion, we tou upon neessry round in reversile nd quntum iruits eders milir wit quntum iruits my inore tis setion Setion 3 summrizes te previous wor on quntum nd reversile iruit syntesis In Setion 4, te proposed rottion-sed tenique is desried In Setion 5, we provide n etension o te proposed syntesis loritm to ndle more enerl loi untions, ie, untions wit inry inputs nd ritrry outputs Syntesis o severl untion milies re disussed in Setion 6, nd inlly Setion 7 onludes te pper A prtil version o tis pper ws presented in [5] Bsi Conepts A quntum it, quit, n e relized y pysil system su s poton, n eletron or n ion In tis pper, we tret quit s mtemtil ojet wi represents quntum stte wit two si sttes 0 nd A quit n et ny liner omintion o its si sttes, lled superposition, s ψ = α 0 + β were α nd β re omple numers nd α + β = Altou quit n et ny liner omintion o its si sttes, wen quit is mesured, its stte ollpses into te sis 0 nd wit te proility o α nd β, respetively It is lso ommon to denote te stte o sinle quit y vetor s [ α β ] T in Hilert spe H were supersript T stnds or te trnspose o vetor A quntum system wi ontins n quits is oten lled quntum reister o size n Aordinly, n n-quit quntum reister n e desried y n element ψ = ψ ψ ψ n simply ψ ψ ψ n in te tensor produt Hilert spe H = H H H n An n-quit quntum te perorms speii n n unitry opertion on seleted n quits A mtri is unitry i = I were is te onjute trnspose o nd I is te identity mtri Te unitry mtri implemented y severl tes tin on dierent quits independently n e lulted y te tensor produt o teir mtries Two or more quntum tes n e sded to onstrut quntum iruit For set o tes,,, sded in quntum iruit C in sequene, te mtri o C n e lulted s M M M were M i is te mtri o te i t te i For quntum iruit wit unitry mtri nd input vetor ψ, te output vetor is ψ = ψ

3 A Adolli, M Seedi, M Pedrm 3 p = p = p = q = q = q = r = r = Fi CNOT nd Tooli tes Deomposition o Tooli te into -quit tes were = ii + ix/ [3] rious quntum tes wit dierent untionlities ve een introdued Te θ-rottion tes 0 θ round te, y nd z es tin on one quit re deined s Eq Te sinle-quit NOT te is desried y te mtri X in Eq Te CNOT ontrolled NOT ts on two quits ontrol nd tret is desried y te mtri representtion sown in Eq Te Hdmrd te, H, s te mtri representtion sown in Eq θ = os θ i sin θ i sin θ os θ os θ sin θ, yθ = sin θ os θ e iθ 0, zθ = 0 e iθ X = 0 0, CNOT = , H = Given ny unitry over m quits m, ontrolled- te wit ontrol quits y y y my e deined s n m + -quit te tt pplies on m i y y y = For emple, CNOT is te ontrolled-not wit sinle ontrol, Tooli is NOT te wit two ontrols, nd C θ is θ te wit sinle ontrol Similrly, multiple-ontrol Tooli te C NOT is NOT te wit ontrols Fi sows CNOT nd Tooli tes For iruit implementin unitry, it is possile to implement iruit or te ontrolled- opertion y replin every te y ontrolled te In iruit dirms, is used or onditionin on te quit ein set to vlue one 3 Previous Wor Syntesis o 0- unitry mtries, lso lled permuttion, s een ollowed y severl reserers durin te reent yers Here, we review te reent pproes wit vorle results More inormtion n e ound in [7] Trnsormtion-sed metods [6] itertively selet te to me iven untion more similr to te identity untion Tese metods onstrut ompt iruits minly or permuttions wit repetin ptterns in output odewords Ser-sed metods [7] eplore ser tree to ind reliztion Tese metods re ily useul i te numer o iruit lines nd te numer o tes in te inl iruit re smll Cyle-sed metods [8] deompose iven permuttion into set o disjoint oten smll yles nd syntesize individul yles seprtely Tese metods re minly eiient or permuttions witout repetin output ptterns BDD-sed metods [9] use inry deision dirms to improve srin etween ontrols o reversile tes Tese teniques sle etter tn oters However, tey require lre numer o nill quits Quntum-loi syntesis dels wit enerl unitry mtries nd is more llenin tn reversile-loi syntesis Syntesis o n ritrry unitry mtri rom universl set o

4 4 eversile Loi Syntesis y Quntum ottion Gtes p = p = q = q = r r = / / p q r p = q = / r = Fi New deinitions or CNOT nd Tooli tes usin ontrolled rottion tes Deomposition o Tooli te into 5 -quit ontrolled-rottion tes tes inludin one-quit opertions nd CNOTs s ri istory Breno et l in 995 [3] sowed tt te numer o CNOT tes required to implement n ritrry unitry mtri over n quits ws On 3 4 n As o 0, te most ompt iruit onstrutions use n 3 n CNOTs [8, 0] nd 4n + n n one-quit tes [] Te srpest lower ound on te numer o CNOT tes is 4 4n 3n [] Dierent trde-os etween te numer o one-quit tes nd CNOTs re eplored in [3] 4 ottion-bsed Syntesis o Boolen Funtions In tis setion, we ddress te prolem o utomtilly syntesizin iven Boolen untion y usin rottion nd ontrolled-rottion tes round te is In tis pper, we ne te sis sttes s ˆ0 = [ 0 ] T nd ˆ = ˆ0 = [ 0 i ] T Wit tis deinition o ˆ0 nd ˆ, te sis sttes remin ortoonl Also, inversion ie, te NOT te rom one sis stte to te oter is simply otined y te Susequently, te CNOT te n e desried y usin te C opertor sown in Fi In ddition, te Tooli te my e desried y usin te C opertor illustrted in Fi Tooli te n e implemented usin 5 ontrolled-rottion opertors s demonstrted in Fi ell tt 3-quit Tooli te needs 5 -quit tes i 0 nd re used s te sis sttes Fi For -quit C θ te wit ontrol quit nd tret quit, te irst output is equl to However, te seond output depends on ot te ontrol line nd te tret line We use te nottion θ to desrie te seond output Furtermore, we write θ to unonditionlly pply sinle-quit θ to te quit Additionlly, one n sow tt or inry vriles,, we ve θ [ θ ] = θ + θ, θ [ θ ] = θ [ θ ], ˆ = is used or netion, nd ˆ0 = Deinition ˆ0 nd ll vriles re in te rottion-sed tored tored in sort orm I nd re in te tored orm, ten θ nd θ re in te tored orm too In quntum iruit syntesized wit θ nd C θ opertors, ll outputs nd intermedite sinls in te iven iruit n e desried in te tored orm For emple, te output r in Fi n e desried s r = [ ] / [ / [ /]] Deinition ˆ0 nd ll vriles re rottion-sed sde sde in sort epressions Wile we used ˆ0 nd ˆ s te sis sttes, te presented loritm n e esily modiied to e pplile to quntum untions desried in terms o 0 nd An lternte solution is to deine te ollowin opertors nd use M to trnsorm te 0 nd sttes to ˆ0 nd ˆ sttes nd opertor M to perorm te reverse trnsormtion Hene to ompute in 0 nd sis, one needs to pply M nd M sinle-quit opertors eore nd ter te omputtion done in te ˆ0 nd ˆ sis, respetively Notie tt M nd M re rottions round te z is M = [ 0 0 i ], M = [ 0 0 i ]

5 A Adolli, M Seedi, M Pedrm 5 I is sde epression nd v is vrile, ten θ nd v θ re sde epressions too θ A sde epression n e epressed s θ 0 [ v θ [ v θ [v n θ n ˆ0 ] ] ] Te prolem o relizin untion wit θ nd C θ opertors is equivlent to indin sde epression or te untion To do tis, we irst introdue rp-sed dt struture in te orm o deision dirm or representin untions 4 A ottion-bsed Dt Struture Te onept o inry deision dirm BDD ws irst proposed y Lee [4] nd lter developed y Aers [5] nd ten y Brynt [6], wo introdued edued Ordered BDD OBDD nd proved its noniity property Brynt lso provided set o opertors or mnipultin OBDDs In tis pper, we omit te prei O BDD s een etensively used in lssil loi syntesis Furtermore, severl vrints o BDD were lso proposed or loi syntesis [9], veriition [7, 8, 9] nd simultion [30, 3] o reversile nd quntum iruits In tis setion, we desrie new deision dirm or te representtion o untions sed on rottion opertors Net, we use it to propose syntesis rmewor or loi syntesis wit rottion tes Deinition 3 A ottion-sed Deision Dirm DD is direted yli rp wit tree types o nodes: sinle terminl node wit vlue ˆ0, weited root node, nd set o non-terminl internl nodes E internl node represents untion nd is ssoited wit inry deision vrile wit two outoin edes: weited ˆ-ede solid line ledin to noter node, te ˆ-ild, nd non-weited ˆ0-ede dsed line ledin to noter node, te ˆ0-ild Te weits o te root node nd ˆ-edes re in te orm o θ mtries We ssume tt < θ Wen weit eiter or n ede or te root node is te identity mtri ie, 0 = I, it is not sown in te dirm Te let DD in Fi 3 sows n internl node wit deision vrile, te orrespondin ˆ0 nd ˆ edes, nd ild nodes 0 nd Te reltion etween te DD nodes in tis iure is s ollows I = ˆ, ten = θ else = 0 In ddition, i is weited root node s sown in te rit DD in Fi 3, ten or = ˆ we ve = θ r θ = θ r + θ ; oterwise = θ r 0 Similr to BDDs, in DDs isomorpi su-rps wi re nodes wit te sme untions re mered Additionlly, i te ˆ0-ild nd te ˆ-ild o node re te sme nd te weit o ˆ-ede is 0 = I, ten tt node is eliminted sin tese two redution rules nd iven totl orderin on input vriles, one n uniquely onstrut te DD o iven untion Notly, deision dirm lled DDMF ws proposed in [8], were e ede n represent ny unitry mtri inludin rottion opertors DDMF ws used or veriition o quntum iruits For iven untion wit n inry vriles v, v,, v n, e vlue ssinment to v, v,, v n orresponds to pt rom te root to te terminl node in te DD o Assumin te vrile orderin v < v < < v n, te orrespondin pt n e identiied y top-down trversl o te DD strtin rom te root node For e node visited durin te trversl, we selet te ede orrespondin to te vlue o its deision vrile v i Denote te weit o te root node y w 0 nd te weit o te seleted [ edes ] y T w, w,, w n We ve v, v,, v n = w 0 w w n ˆ0 = w 0 w w n 0 I ˆ0-ede is seleted or vrile v i ie, i v i = ˆ0, we ve w i = I Note tt wen te

6 Clerly, i durin tis rp trversl 0 -ede is seleted or vrile v i ie, i v i = 0, ten te orrespondin ede weit will e w i =I [ r + ] [ r 0 ] [ r + ] [ r We ve sown tt QDD s provide onise nd nonil representtion or quntum ule : < Consider perormin q-pply to to otin q-pply tes two two QDD nodes nd nd s s untions Notie tt QDD s n e rerded s enerliztion o BDD s ie, e BDD n lso ruments e rerded nd nd s ompres QDD te te A orrespondin QDD is BDD deision etly vriles i ll o te o te te weits nodes o Net, te ter QDD ter inludin re eiter [ r + ] [ r + ] [ r 0 ] [ r 0 ] te te weits o 6o te te eversile root root node node nd Loi nd ˆ Syntesis ˆ -ede in in te te y Quntum orrespondin ottion ˆ ˆ -ild Gtes nd nd -ild, it it dds dds new new 0 I or As will e sown lter, te syntesis proess strts wit te QDD o te node node to to te te resultin QDD,,, y y usin one one o o tree tree rules rules provided in in Fiure 7 7 Assume tt tt te te ule 3: = iven loi untion wi is lso QDD nd deomposes te iven QDD to relizle = orrespondin vriles or or QDD nodes r nd nd re re nd nd,, respetively Fiure Te 7 Te ules new new or node implementin node te q-pply opertor on two QDD s QDD s enerted y y q-pply depends on on te te vrile orderin o o nd nd s s demonstrted = in in Fiure 7 7 For terminl onditions, te ollowin reltions re used: v v nd ˆ v v Te QDD struture For For emple, suppose s some useul tt tt < < 0 ule ule properties is is invoed, One importnt 0 enertin new new node node in in te te resultin QDD Sine only ssumes nd ˆ vlues, tese re te only possile terminl onditions property, ie, te liner topoloy property, is demonstrted in Fiure θ = θ ontinin vrile ule ule direts te te q-pply opertion to to reursively ll ll itsel Ater reursive omputtion o te ˆ -ild nd -ild o, in order to mintin te noniit 5 Te ide is tt wen te -ild nd te ˆ -ild o node re Fiure 5: sd o te resultin QDD, isomorpi su-rps re mered nd i te -ild nd te ˆ -ild o te sme node Fi, 3 ten tt Internl node struture n e o diretly rottion-sed relized y deision dirm DD witout nd wit weited Fiure 5 Te liner r r root For node, i te ˆ0-ild nd tenode ˆ-ild re te resme tetopoloy sme nd node weit property, o te n o ˆ e -ede diretly is relized y ontrolled- θ opertor opertor in terms s sown o its in ild ie, QDD r r r r 0 I, ten tt node will v v 0 e eliminted In q ddition, me rqdd o nonil, te resultin 0 0 w w w 0 w 0 weits or te nodes ˆ -ild nd -ild o [r+] re modiied / s r demonstrted in Fiure 8 to ule : < q-pply tes two two QDD nodes nd nd s s Fiure 6 [ QDD s [ r + r or + ] intermedite ] [ [ r + sinls r + ] ] o te syntesized [ [ r r 0 ] 0 tree-input ] [ [ r Tooli r 0 ] 0 ] te Consider perormin q-pply to to otin ruments nd nd ompres te te orrespondin deision vriles o o te te nodes Net, ter ter inludin ule : < Fiure 7 7 ules or or implementin te te q-pply opertor on on two two QDD s ˆ0-ildmultiplyin nd te ˆ-ild te o root nodeweit re te o sme ynode γ, ten re modiied Tott otin s demonstrted node n in Fiure = e 8 to diretly γ relized or iven DDs o use te pply opertor Fiure 8 Weit modiition durin q-pply to mintin noniity o te resultin QDD y θ opertor, s = θindemonstrted tis ontet, infi nd3 re ndlled Fi 3, DD in terms opernds o its Te ppl ild Fi is implemented 4 sows te ydds reursive Fiure o untions 6: trversl Opernds p, qo or nd te opertion rtwo in Fi DD = γ opernds reprodued Te result For in o pply Fi eopertor pir o wi nodes dds i [r+] [r0] 6 4 visited Every DD durin wite trversl, innew struture node ntointernl te su resultin s te node DD ones is dded sown y usin in one tofi o te4 resultin treeisrules: ssoited i DD < ule y, utilizin v =, w te = [ rules wi depend on te α seleted r + α ] γ, w vrile 0 = [ orderin α r 0 ] γ I < ule, v =, w lso see Fiure 6 We = ssume γ[ β tt r + β ], wit sde epression nd nw e 0 = relized γ[ wit β r 0 rottion ] I = nd ule ontrolled-rottion 6 3, 6v = =, w = opertors [ α r + α ] γ[ β r + β ], Suppose two enerl tt tedds or sown w 0 untion = [α in Fiure r 0 ] isγ[ 6 iven β r 0 Te ] Weit pply DDmodiition or opertor = or te isγ pply reursively n opertor e tolled mintinwit noniity t o te resultin DD otined onditions y multiplyin ˆ0 θv te= root v[r+] nd weit ˆ [r+] o θv y = γ [r0] θv To otin [r0] = γ or iven DDs o nd, we use te pply opertor In tis ontet, nd re lled DD 4 Opertions on DDs opernds o Te pply opertor is implemented y reursive trversl o te two DD Suppose tt te DD or untion is iven Te DD or = γ n e otined y opernds For e ule pir o nodes Fiure < 7 7 ules Te or or implementin newte node te q-pply or opertor on two two is QDD s multiplyin in ndte visited root weit durin o yte γ trversl, To Te otin weits n = internl γ o ˆ-ild node or iven is DDs nd ˆ0-ild o nd, re we dded to te resultin α ]DD γ, ynd use utilizin te [pply αte opertor r ollowin 0 ] γ, In tisrules ontet, respetively wi nd depend re lledondd teopernds seletedo Te pply opertor is implemented y reursive trversl o te two DD opernds For e pir o nodes in nd vrile orderin lso see Fi visited 5 durin We te ssume trversl, tt n internl ndnode ve is dded two to te enerl resultindds y utilizin te ollowin sown in Fi 5 ule Te pply < rules opertor Te wi new depend is node reursively on te or seleted lled is vrile Te wit orderin weits te terminl lso osee ˆ-ild Fiure onditions 6 nd We ssume ˆ0-ild ttre nd ve γ 6 6 β ], nd two γ[ enerl β DDs r 0 ], sown respetively in Fiure 6 Te pply opertor is reursively lled wit te terminl ˆ0 θv = v nd ˆ θv = θv onditions ˆ0 θv = v nd ˆ θv = θv 0-0 Fiure 8 Weit modiition durin q ule : < to mintin noniity o te resultin QD / [r+] [r+] [r0] [r0] p Fiure te te weits o 6: Opernds o te te root root node node nd nd ˆ ˆ -ede in in te te orrespondin or opertion ˆ ˆ -ild nd nd -ild, it it dds dds new new = γ Te result o pply opertor w node node to to te te [ [ r + r + ] ] [ [ r r 0 ] 0 ] new node to te resultin QDD,,, y y usin one one o o tree tree rules DD y rules provided in in Fiure 7 7 Assume tt tt te te ule 3: = usin one o te tree rules: i < ule, v = orrespondin vriles or or QDD nodes nd nd re re nd nd,, respetively Fiure Te 7 Te ules new new or node implementin node te q-pply opertor on two QDD s [ α r + α ] γ, w 0 = [ α r 0 ] γ I < ule, v = p, w = γ[ β enerted y y q-pply depends on on te te vrile orderin o o nd For nd terminl s s demonstrted onditions, in te in ollowin Fiure 7 7 reltions re used: w 0 = γ[ β r 0 ] I = ule 3, v = =, w = [ q α v v nd ˆ v v r + α ] γ[ β For For emple, suppose tt tt < < ule ule is is invoed, enertin new new node node in in te te resultin r QDD Sine only ssumes nd ˆ vlues, tese re te only possile terminl onditions w 0 = [α r 0 ] γ[ β r 0 ] Weit modiition / / or te pply r opertor to mintin ontinin vrile ule ule direts te te q-pply opertion to to reursively ll ll itsel o te resultin DD Ater reursive omputtion o te ˆ -ild nd -ild o, in order to mintin te noniity Fi 4 DDs or intermedite sinls o 3-input Tooli te sown in Fi, redrwn in In tis r r I v v 4 Opertions on DDs 0, ten tt node will e eliminted In iure, we ve q =, r = /r, nd r = / Suppose tt te DD or untion is iven 0 0 [r0] Fiure 5: sd o te resultin QDD, isomorpi su-rps re mered nd i te -ild nd te ˆ -ild o node re te sme nd te weit o te ˆ -ede is ddition, me QDD o nonil, te resultin w w weits or te nodes ˆ -ild nd -ild o Opernds or opertion = [r+] Te DD w0 w0 or = γ n e o ule 3 = Te new node or is or Te weits o ˆ-ild nd ˆ0 [ α r + α ] γ[ ule β r + < β Te ], new nd node [α or is r Te 0 ] weits γ[o β ˆ-ild r nd 0 ], respetively ˆ0-ild re [ α r + In enerl, or inry opertion op nd two BDDs o untions nd, te pply opertor omputes BDD or op [6] α ] γ, nd [ α r 0 ] γ, respetively Ater reursive omputtionβ o ], te nd ˆ-ild γ[ βnd r 0 ], ˆ0-ild respetively o, to mintin te noniity o t DD, isomorpi su-rps ule re3 mered = Te nd newi node teorˆ0-ild is ornd Te teweits ˆ-ild o ˆ-ild o ndnode ˆ0-ild re [ nd te weit o te ˆ-ede is α r + α 0 ] = γ[ I, ten β r + β tt ], nd [α node r will 0 ] γ[ e eliminted β r 0 ], respetively In ddition DD o nonil, te resultin weits or te ˆ-ild nd te ˆ0-ild o sould e m Ater reursive omputtion o te ˆ-ild nd ˆ0-ild o, to mintin te noniity o te resultin te metod illustrted indd, Fiure isomorpi 6 su-rps Fiure re 7 mered demonstrtes nd i te ˆ0-ild te ndresult te ˆ-ild o perormin o node re te ppl sme nd te weit o te ˆ-ede is on q nd r in Fiure 4 redrwn in Fiure 0 = I, ten tt node will e eliminted In ddition, 7 to otin r = q /r 3 to me To 0 [r0] 0-0 ule < Te new node or is Te weits o ˆ-ild nd ˆ0-ild re γ[ β r +

7 0 0 Opernds or opertion = [ r+ ] [ r 0] [ r+ ] [ r 0] ule : < Consider perormin q-pply q-pply to otin to otin q-pply q-pply tes tes two two QDD QDD nodes nodes nd nd s s ruments nd nd ompres te te orrespondin deision vriles o te o te nodes nodes Net, Net, ter ter inludin [ r+ ] [ r+ ] te te weits o te o te root root node node nd nd ˆ -ede ˆ -ede in te in te orrespondin ˆ -ild ˆ -ild nd nd -ild, -ild, it dds it dds new new ule : < A Adolli, M Seedi, M Pedrm 7 ule < Te new node or is Te weits o ˆ-ild nd ˆ0-ild re ule 3: = [ α r + α ] γ, nd [ α r 0 ] γ, respetively node node to te to te resultin QDD, QDD,, y, y usin usin one one o tree o tree rules rules provided in in Fiure Fiure 7 7 Assume tt tt te te [ r 0] [ r 0] orrespondin vriles or or QDD QDD nodes nodes nd nd re re nd nd,, respetively Fiure Te 7 Te ules new new or node implementin node te q-pply opertor on two QDD s enerted y y q-pply q-pply depends on on te te vrile orderin o o nd nd s s demonstrted in Fiure in Fiure 7 7 ule < Te new node or is Te weits o ˆ-ild nd ˆ0-ild re For terminl onditions, te ollowin reltions re used: v v γ[ β r + β ], nd γ[ β r 0 ], respetively nd ˆ v v For For emple, suppose tt tt < < ule ule is is invoed, enertin new new node node in te in te resultin QDD QDD Sine only ssumes nd ˆ vlues, tese re te only possile terminl onditions ontinin vrile ule ule direts direts te te q-pply q-pply opertion to to reursively ll ll itsel itsel ule 3 = Te new node or is or Te weits o ˆ-ild nd ˆ0-ild re [ α r + α ] Ater γ[ reursive βomputtion r + β o ], te nd ˆ -ild [α nd -ild r 0 o ], in γ[ order to mintin β r te 0 ], noniity respetively o te resultin QDD, isomorpi su-rps re mered nd i te -ild nd te ˆ -ild o r r node re te sme nd te weit o te ˆ -ede is r r 0 I, ten tt node vwill v e eliminted In 0 ddition, me QDD o nonil, te resultin w w w 0 w 0 weits or te nodes ˆ -ild nd -ild o re modiied s demonstrted in Fiure 8 to 0 Fiure 8 Weit modiition durin q-pply to mintin noniity o te resultin QDD Fi 5 Opernds or opertion = γ Te result o pply opertor wi dds new node to te resultin DD y usin [ [ r+ one r+ ] o ] te tree rules: [ [ i r 0] r < 0] ule, v =, w 6 = [ α r + α ] γ, w 0 = [ α r 0 ] γ I < ule, v =, w = γ[ β r + β ], w 0 = γ[ β r 0 ] I = ule 3, v = =, w = [ α r + α ] γ[ β r + β ], w 0 = [α r 0 ] γ[ β r 0 ] Weit modiition or te pply opertor to mintin noniity o te resultin DD [ [ r+ r+ ] ] [ [ r+ r+ ] ] [ [ r 0] r 0] [ [ r 0] r 0] Ater reursive omputtion o te ˆ-ild nd ˆ0-ild o, to mintin te noniity o te resultin DD, isomorpi su-rps re mered nd i te ˆ0-ild nd te ˆ-ild o node re te sme nd te weit o te ˆ-ede is 0 = I, ten tt node will e eliminted In ddition, to me DD o nonil, te resultin weits or te ˆ-ild nd te Fiure Fiure 7 ules 7 ules or or implementin te te q-pply q-pply opertor on two on two QDD s QDD s ˆ0-ild o sould e modiied y te metod illustrted in Fi 5 Fi 6 demonstrtes te result o perormin pply opertor on q nd r in Fi 4, redrwn in Fi 6, to otin r = q /r To onstrut DD or r, one needs to initilly pply ule 3 euse ot q nd r use s roots Aordinly, 6 6 w = [ ] /[ /r ] nd w 0 = /r To ontinue, onsider w nd note tt ot [ ] nd [ /r ] use 3 As result, pplyin ule 3 leds to w, = [ 0] /[ / + /] nd w,0 = [ ˆ0] /[ /] On te oter nd, pplyin ule 3 on w 0 leds to w 0, = [ ˆ0] /[ /] nd w 0,0 = [ 0] / sin terminl onditions results in w, =, w,0 =, w 0, =, nd w 0,0 = Sine w 0, = w 0,0 =, we n remove vrile s te ˆ0-ild o Te inl iure in Fi 6 is otined ter elimintin redundnt nodes nd edes 4 4 Funtionl Deomposition nd r-linerity Te prolem o relizin untion usin θ nd C θ opertors is equivlent to indin rottion-sed tored orm or, wi n e perormed y reursive i-deomposition 3 To understnd te DDs o [ ] nd [ /r ], rell DDs o nd r in Fi 6 nd use weits nd / or roots o nd r, respetively 4 Note tt te ommuttive property o mtri multiplition or θ mtries is ritil or te pply opertor Perormin pply s desried my not enerte te orret result or deision dirms wit non-ommuttive weits

8 re modiied s demonstrted in Fiure 8 to Fiure 9 demonstrtes te result o perormin q-pply opertion on q nd r ten rom Fiure 6 to otin r=q -/r It is noteworty tt te ommuttive property o mtri multiplition or mtries is ritil i tere or eists te rottion q-pply vlue, to enerte i te orret result ie, q / / In te reminder o tis pper, wen epressin untion s v, v,, v, it is impliitly r Net we present numer o ey results r = q -/ r deision vrile v i Te weit o te ˆ -ede o n i will e i v n Also no ede oriintin rom nodes ove n j ie, nodes wit n deision vrile v j, j<i will end t node elow n i ie, node wit deision vrile v j, j>i Fiure 9 An emple o perormin q-pply on two QDD s 44 QDD-sed Funtionl Deomposition nd te Notion o Q-Linerity Fiure 0 A enerl QDD Fi 6 An emple o perormin pply opertor on two DDs In te irst ll o te struture pplywit opertor, q-liner As mentioned erlier, te prolem o relizin untion,, usin nd ontrolled- vriles v+ w vn = [ ] /[ /r ] nd w 0 = /r Te inl iure is otined ter elimintin8 redundnt nodes nd edes A enerl DD struture wit r-liner vriles v +,, v n opertors is equivlent to indin quntum tored orm or te untion, wi n in turn e perormed y reursive i-deomposition o We reer te reder to Li et l [37] nd Krplus o [38] or review o prior wor relted to untionl deomposition in enerl, nd ideomposition untions in prtiulr nd nd vlue γ su tt = γ Deinition 4 ottion-sed i-deomposition i-deomposition in sort o is deined s indin Deinition: We usequntum i-omposition i-deomposition o iven o is untion deined s indin to onstrut untions nd = nd γ vlue Susequently, nd re reursively i-deomposed, wi will eventully result in tored orm o Te i-deomposition su tt loritm were only is ssumes sed on vlues te notion nd ˆ o r-linerity n ssumed tt depends on ll vriles v, v,, vn ie, is not invrint wit respet to ny vrile mon v, v,, vn Deinition: For iven untion v, v,, v, v, v,, v, vrile v i will e lled q-liner v,,, v,, vi, vi vn vlues, i i i n, su tt or every vlue ssinment to v v,, v, v,, v, ten or e vrile v i, +<i<n, tere is only one QDD node, n i, wit vi i vi, i i n : i, were we deine perormin q-pply s desried my not v i vi v v, v,, vi,ˆ, vi,, vn nd enerte te orret result or deision dirms wit 8 eversile Loi Syntesis y Quntum ottion Gtes v, v,, vi,, vi,, vn v A vrile will e lled q- weits tt re not ommuttive nonliner i it is not q-liner v Lemm : Consider untion v, v,, vn wit vrile orderin v r v v n I nd only i vriles v, v,, vn re q- liner ie, or e v i, +<i<n, tere is i tt or ll Deinition 5 For untion v,, v i, v i, v i+,, v n, vrile v i is r-liner i tere eists rottion vlue θ i su tt or every vlue ssinment to v,, v i, v i+,, v n : vi = θ i vi, were vi = v,, v i, ˆ, v i+,, v n nd vi = v,, v i, ˆ0, v i+,, v n A vrile is r-nonliner i it is not r-liner 7 Now we present numer o ey results v + Lemm Consider untion v, v,, v n wit vrile orderin v < v < < v n nd ssume tt + i n I e vrile v i is r-liner, ten tere is only one DD node n i or e r-liner deision vrile v i Te weit o te ˆ-ede o n i is θ i Proo Te proo is y indution on v n, v n, v n,, v + strtin rom v n Let v e te lowest indeed r-nonliner vrile ter wi v +, v +,, v n re r-liner vriles o From Lemm, or + j n we ve vj = θ j vj were θ j is ied independent o v, v,, v j, v j+,, v n vlues As illustrted in Fi 6, every pt rom te root node o te DD to te terminl node will eiter o trou n internl node wit deision vrile v or it will sip ny su node nd diretly o te sinle DD node wit deision vrile v + For te ltter se, v = 0 v = v nd or ny ormer se v = α i v or some vs ll v,, v, v +,, v n Additionlly, te numer o dierent rottion nles e, α, α in Fi 6 or vrile v is equl to te numer o internl nodes wit deision vrile v in te DD v+ v+ v

9 A Adolli, M Seedi, M Pedrm 9 Aloritm tor : I ll vriles re r-liner, ten return te orrespondin sde epression or : Find te lowest indeed r-nonliner vrile v ter wi v +, v +,, v n re r-liner 3: Bi-deompose usin v s = γ were,, nd γ re iven in Teorem 4: eturn [tor ] γ[tor] Deinition 6 Te deree o r-nonlinerity o vrile v, r-dev i, is m were m is te numer o dierent rottion nles α i inludin 0 i ny tt v = α i v or some v,, v, v +,, v n For r-liner vriles te deree o r-nonlinerity is zero As n emple, onsider te DD o r in Fi 6 nd note tt r-de = s tere re two rottion nles ie, 0 nd or Similrly, r-de = 0 nd is r-liner Lemm Let m denote te numer o internl nodes wit deision vrile v I ll pts rom te root node o te DD to te terminl node o trou n internl node wit deision vrile v, r-deqv = m ; oterwise r-dev = m Proo Te proo ollows rom onsiderin te enerl struture o DDs nd te deinition o r-nonlinerity Teorem Consider untion v, v,, v n wit vrile orderin v < v < < v n Deine su tt i v = α v ten = ˆ; oterwise = ˆ0 Assume tt v +, v +,, v n re r-liner vriles o nd v is r-nonliner vrile o wit r- dev = m Additionlly, or e vlue ssinment to vriles v,, v, v +,, v n suppose etly one o te ollowin m reltions olds: v = α v, v = α v,, v = α m v We ve I n e i-deomposed s = γ were = v, γ = α α /, = γ II is untion o v, v,, v, ie, is invrint wit respet to v +, v +,, v n III v is r-liner vrile o I is untion o v, v,, v n nd v +, v +,, v n re r-liner vriles o r-dev in is m Proo We initilly prove tt untion is invrint wit respet to v +, v +,, v n, ie, vi = vi or + i n Sine v i is r-liner, tere eists θ i su tt or ll v,, v i, v i+,, v n vlues, vi = θ i vi wi results in viv = θ i viv nd viv = θ i viv From te deinition o we ve: I viv = α viv, ten vi = ˆ, else vi = ˆ0 I viv = α viv, ten vi = ˆ, else vi = ˆ0 Cominin tese reltions proves vi = vi : viv = α viv θ i viv = α + θ i viv viv = α viv

10 0 eversile Loi Syntesis y Quntum ottion Gtes Sine = v, is lso invrint wit respet to v +, v +,, v n prt II Moreover v = nd v = wi results in v = v, ie, v in is r-liner prt III Te irst sentene o prt I is ler rom te deinition o = γ As or te seond one, note tt v +, v +,, v n re r-liner vriles o Additionlly, is invrint wit respet to v +, v +,, v n Puttin tese ts toeter proves prt I Now we prove r-dev m in = γ For e vlue ssinment to vriles v, v,, v, v +,, v n etly one o te ollowin m reltions olds: v = α v, v = α v,, v = α m v For e o te ove ses, we emine te reltion etween v nd v : v = α v : By deinition = ˆ nd we ve: v = ˆ γ v = γ v v = γ v v = [ ˆ] γ v = v = α v = α + γ v, γ = α α / v = α+α v v = α v : By deinition = ˆ0 nd we ve: v = ˆ0 γ v = v v = [ ˆ0] γ v = γ v = γ + α v, γ = α α / v = α+α v v = α i v, 3 i m: By deinition = ˆ0 nd v = γ + α i v Te irst two ses result in te sme reltion etween v nd v s v = α+α v Te reminin m ses result in t most m dierent reltions etween v nd v Tereore, te totl numer o dierent reltions etween v nd v is m Aordinly, r-dev in is m prt Finlly, rom = γ it ollows tt γ = γ [ γ] Consider = v nd ssume = ˆ or = ˆ0 wi leds to = v ˆ = v or = v ˆ0 = v Altoeter or ot v = ˆ nd v = ˆ0, we ve γ [ γ] = Hene, n e i-deomposed s = γ prt I sin te proposed i-deomposition ppro, n e i-deomposed into = γ were nd re temselves reursively i-deomposed until rottion-sed tored orm is otined Teorem Te proposed i-deomposition ppro lwys results in sde epression or iven untion Proo Followin te deinitions iven in Teorem or = γ, sine is invrint o v +, v +,, v n nd v in is r-liner nd r-dev in is m, te reursion will inlly stop t terminl ses were nd/or ve diretly relizle DDs ll vriles re r-liner in te untions nd tey ve rottion-sed sde epressions orrespondin to DDs wit in struture 5 5 As result o Lemm, in untion wit in strutured DD, ll vriles re r-liner

11 A Adolli, M Seedi, M Pedrm Aloritm -tordd : Cne ll o te weits to 0 = I : Crete DD node v wit w = nd w 0 = I to te terminl node ie, ˆ0 3: ediret ll edes towrd n to node v nd me te weit o ll su edes 4: ediret ll edes towrd n, n 3,, n m to node v nd me te weit o ll su edes 0 5: Disrd nodes n, n 3,, n m 6: Mere isomorpi su-rps, eliminte nodes wit te sme ˆ0-ild nd ˆ-ild etly i te weit o te ˆ-ede is 0 = I pdte weits o te DD to me te DD o nonil Aloritm uses te proposed reursive i-deomposition ppro to enerte rottionsed tored orm or iven untion All steps in Aloritm n e diretly perormed on DDs I te DD o untion is in struture, we ve sde epression or Step For Step s depited in Fi 6 nd ordin to Lemm, identiyin v is equivlent to identiyin te lower in-struture prt o te DD As or Step 3, ordin to Lemm vlues α, α,, α m n e otined rom weits o te ˆ-edes o nodes wit deision vriles v Hene, γ = α α / is otined Let n i i m denote nodes wit deision vrile v nd ˆ-edes weit α i Strtin rom te DD o, one n perorm Aloritm to onstrut DD o Hvin te DDs or nd, te DD o = γ n e otined y usin te pply opertion As n emple o Aloritm, see DDs o s nd in Fi 8 nd Fi 9 were v = nd m = Tis emple is desried in detil in Setion 6 Te inl orm ter pply is = γ [ γ [ 3 γ 3 [ γ ˆ0 ] ] ] Note tt i untions sould lso e deomposed Te tor loritm is not optiml In prtiulr, n e rewritten s = p γ p [ p γ p [ p3 γ p3 [ p γ p ˆ0 ] ] ] were p, p,, p is permuttion o,,, Dierent permuttions o,,, my result in dierent numer o tes ter syntesis For emple, onsider te DDs o te output s in 4-input Tooli te, sown in Fi 8, or two dierent vrile orderins > > > d nd > > d > In Fi 8, d is r-liner However, none o te vriles in Fi 8 re r-liner Aordinly, te proposed ppro results in ewer tes or > > > d Te ormer se is urter disussed in Setion 6 Indeed, worin wit > > d > leds to s = / were = d is 4-input Tooli te treted on te lst quit or > > d > 5 Worin wit Aritrry Outputs For te input vetor, untion wit inry inputs nd outputs n e written s = ĝ γ [ ĝ γ [ [ĝ γ ˆ0 ] ] ] Sine untions ĝ i only te vlues ˆ0 nd ˆ, n lso e represented s = γ + γ + + γ ˆ0 were i vlues re eiter zero 0 or one 6 Deine γ = γ + γ + + γ wi leds to = γˆ0 Aordinly, te struture o te syntesized 6 To prove, ssin ritrry vlues ˆ0 nd ˆ to ĝ i terms nd onsider te resultin rottions y dierent γ i vlues

12 eversile Loi Syntesis y Quntum ottion Gtes \ G G ˆ0 \ γ Fiure : sdsd 8 tm n esily e enerlized to e pplile to m untions wose output vlues or ll possile sis vetors re on sinle ritrry irle C wit te oriin enter E point on irle C n e represented s ˆ were is vetor pssin tru te oriin nd diulr to irle C In ddition, is te quntum - rottion opertion round te vetor nd ˆ0 n e set point on te irle C Fiure 9 untion n e represented wit QDD were te terminl node is ˆ0 nd te weits o nd te root node re in te orm o Notie tt te quntum iruit syntesized y te q-tor loritm s te property tt or te sis input vetors, te vlues o ll l nd output sinls will lie on some irle, C inl tored orm resultin rom q-pply s te were is te input vetor Sine ons i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso resented s: were i vlues re rerded l vlues 0 nd ˆ Let s deine te rel vlued untion s quently, untion n e represented s Te struture o te syntesized iruit represented s Fiure 0 Note tt G is te inverse on o G wit rerd to X Y ˆ Z C Fiure 9 Blo spere representtion G G Fiure 0 Quntum iruit perormin 8 s its enter E point on irle C n e represented s were is vetor pssin tru te oriin nd perpendiulr to irle C In ddition, is te quntum - deree rottion opertion round te vetor nd ˆ0 n e set to ny point on te irle C Fiure 9 Su untion n e represented wit QDD were te terminl node is ˆ0 nd te weits o edes nd te root node re in te orm o Notie tt te quntum iruit syntesized y usin te q-tor loritm s te property tt or te sis input vetors, te vlues o ll internl nd output sinls will lie on some irle, C Te inl tored orm resultin rom q-pply s te orm were is te input vetor Sine untions i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso e represented s: were i vlues re rerded s rel vlues 0 nd ˆ Let s deine te rel vlued untion s Consequently, untion n e represented s Te struture o te syntesized iruit n e represented s Fiure 0 Note tt G is te inverse untion o G wit rerd to X Y ˆ C Fiure 9 Blo spere representtion Fiure 0 Quntum iruit perormin G G' ' Fiure : Blo spere representtion Quntum iruit perormin = [γ] ˆ0 Te inl tored orm resultin rom q-pply s te orm = γ [ γ [ [ γˆ0 ] ] ] were is te input vetor Sine untions i only te vlues ˆ0 nd ˆ or sis input vetors, it n e seen tt n lso e represented s = [γ + γ + + γ] ˆ0 were i vlues re rerded s rel vlues ˆ0 0 nd ˆ Lets deine te rel vlued untion γ s γ = γ + γ + + γ Consequently, untion n e represented s = [γ] ˆ0 Te struture o te syntesized iruit n e represented s Fiure Note tt G is te inverse untion o G wit rerd to Te rit portion o te iruit is needed only i it is required tt quits ssoited wit input lines mintin teir initil vlue To lriy te roles o G nd G, it will e eneiil to ompre tis iruit wit te tree-input multipleer iruit = s + s syntesized y te q-tor loritm in Fiure 3 I insted o ˆ0, noter quntum vlue q is used in tis iruit s te initil vlue or te input, ten te resultin iruit will implement te ollowin untion = [γ] q Now we enerlize te syntesis ppro or syntesizin quntum untions tt or iven sis input vetors enerte enerl quntum vlue = [ 0 ] T Beuse 0 + =, we my rewrite s = e iδ [ e iγ/ os θ ie iγ/ sin θ ] T Hene, n e epressed s = e iδ zγθˆ0 were z is te rottion opertor round te Z is deined s zγ = [ e iγ/ 0 0 e iγ/ ] We n inore te term e iδ sine it s no oservle eets [] nd tereore we n eetively write = zγθˆ0 see Fiure 4 Tis onept n s s ˆ0 / / / / Fiure 3: Quntum tree-input multipleer 0 8 loritm n esily e enerlized to e pplile to quntum untions wose output vlues or ll possile sis input vetors re on sinle ritrry irle C wit te oriin s its enter E point on irle C n e represented s were is vetor pssin tru te oriin nd perpendiulr to irle C In ddition, is te quntum - deree rottion opertion round te vetor nd ˆ0 n e set to ny point on te irle C Fiure 9 Su untion n e represented wit QDD were te terminl node is ˆ0 nd te weits o edes nd te root node re in te orm o Notie tt te quntum iruit syntesized y usin te q-tor loritm s te property tt or te sis input vetors, te vlues o ll internl nd output sinls will lie on some irle, C Te inl tored orm resultin rom q-pply s te orm were is te input vetor Sine untions i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso e represented s: were i vlues re rerded s rel vlues 0 nd ˆ Let s deine te rel vlued untion s Consequently, untion n e represented s Te struture o te syntesized iruit n e represented s Fiure 0 Note tt G is te inverse untion o G wit rerd to X Y ˆ Z C Fiure 9 Blo spere representtion G G Fiure 0 Quntum iruit perormin \ G G ˆ0 \ γ Fiure : sdsd 8 loritm n esily e enerlized to e pplile to quntum untions wose output vlues or ll possile sis input vetors re on sinle ritrry irle C wit te oriin s its enter E point on irle C n e represented s were is vetor pssin tru te oriin nd perpendiulr to irle C In ddition, is te quntum - deree rottion opertion round te vetor nd ˆ0 n e set to ny point on te irle C Fiure 9 Su untion n e represented wit QDD were te terminl node is ˆ0 nd te weits o edes nd te root node re in te orm o Notie tt te quntum iruit syntesized y usin te q-tor loritm s te property tt or te sis input vetors, te vlues o ll internl nd output sinls will lie on some irle, C Te inl tored orm resultin rom q-pply s te orm were is te input vetor Sine untions i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso e represented s: were i vlues re rerded s rel vlues 0 nd ˆ Let s deine te rel vlued untion s Consequently, untion n e represented s Te struture o te syntesized iruit n e represented s Fiure 0 Note tt G is te inverse untion o G wit rerd to X Y ˆ Z C Fiure 9 Blo spere representtion G G Fiure 0 Quntum iruit perormin 8 usin te q-tor loritm s te property tt or te sis input vetors, te vlues o ll internl nd output sinls will lie on some irle, C Te inl tored orm resultin rom q-pply s te orm were is te input vetor Sine untions i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso e represented s: were i vlues re rerded s rel vlues 0 nd ˆ Let s deine te rel vlued untion s Consequently, untion n e represented s Te struture o te syntesized iruit n e represented s Fiure 0 Note tt G is te inverse untion o G wit rerd to Fiure 0 Quntum iruit perormin G G' ' Fiure : Blo spere representtion Quntum iruit perormin = [γ] ˆ0 Te inl tored orm resultin rom q-pply s te orm = γ [ γ [ [ γ ˆ0 ] ] ] were is te input vetor Sine untions i only te vlues ˆ0 nd ˆ or sis input vetors, it n e seen tt n lso e represented s = [ γ + γ + + γ ] ˆ0 were i vlues re rerded s rel vlues ˆ0 0 nd ˆ Lets deine te rel vlued untion γ s γ = γ + γ + + γ Consequently, untion n e represented s = [γ] ˆ0 Te struture o te syntesized iruit n e represented s Fiure Note tt G is te inverse untion o G wit rerd to Te rit portion o te iruit is needed only i it is required tt quits ssoited wit input lines mintin teir initil vlue To lriy te roles o G nd G, it will e eneiil to ompre tis iruit wit te tree-input multipleer iruit = s + s syntesized y te q-tor loritm in Fiure 3 I insted o ˆ0, noter quntum vlue q is used in tis iruit s te initil vlue or te input, ten te resultin iruit will implement te ollowin untion = [γ] q Now we enerlize te syntesis ppro or syntesizin quntum untions tt or iven sis input vetors enerte enerl quntum vlue = [ 0 ] T Beuse 0 + =, we my rewrite s = e iδ [ e iγ/ os θ ie iγ/ sin θ ] T Hene, n e epressed s = e iδ zγ θˆ0 were z is te rottion opertor round te Z is deined s zγ = [ e iγ/ 0 0 e iγ/ ] We n inore te term e iδ sine it s no oservle eets [] nd tereore we n eetively write = zγ θˆ0 see Fiure 4 Tis onept n s s ˆ0 / / / / Fiure 3: Quntum tree-input multipleer 0 Fiure : Blo spere representtion Quntum iruit perormin = [γ] ˆ0 s s ˆ0 / / / / Fiure : Quntum tree-input multipleer Te inl tored epression s te orm = ĝ γ [ ĝ γ [ [ĝ γ ˆ0 ] ] ] were is te input vetor Sine untions ĝ i only te vlues ˆ0 nd ˆ or sis input vetors, n lso e represented s = γ + γ + + γ ˆ0 were i vlues re eiter zero 0 or one 4 Deine untion γ s γ = γ + γ + + γ wi leds to = γˆ0 Aordinly, te struture o te syntesized iruit n e represented s Fiure In tis iure, G is iruit tt onstruts γ nd G is te inverse o G Note tt G sould e used only i one wnts to eep input lines unned To lriy te roles o G nd G, it will e eneiil to ompre tis iruit wit te tree-input multipleer iruit = s + s syntesized y te q-tor loritm in Fiure I insted o ˆ0, noter quntum vlue q is used in tis iruit s te initil vlue or te input, ten te resultin iruit will implement te ollowin untion = [γ] q Now we enerlize te syntesis ppro or syntesizin quntum untions tt or iven sis input vetors enerte enerl quntum vlue = [ 0 ] T Beuse 0 + =, we my rewrite s = e iδ [ e iγ/ os θ ie iγ/ sin θ ] T Hene, n e epressed s = e iδ zγ θˆ0 were z is te rottion opertor round te Z is deined s zγ = [ e iγ/ 0 0 e iγ/ ] We n inore te term e iδ sine it s no oservle eets [] nd tereore we n eetively write = zγ θˆ0 see Fiure 3 Tis onept n e demonstrted y usin te Blo spere representtion Note tt te quntum vlue zγ θˆ0 results rom θ rottion o ˆ0 round te X is ollowed y γ rottion round te Z is Te quntum iruit or = zγ θˆ0 n e syntesized s Syntesize = θˆ0 y usin te q-tor loritm Syntesize = zγˆ0 y usin te q-tor loritm Csde te resultin iruits s depited in Fiure 3 Te onstnt nill in te riteture o Fiure 3 is not lwys neessry For emple, te ontrolled rottion wit ontrol quit nd tret ˆ0 enertes s te seond output nd te 4 To prove, ssin ritrry vlues ˆ0 nd ˆ to ĝi nd onsider te resultin rottions y dierent γi vlues 0 8 loritm n esily e enerlized to e pplile to quntum untions wose output vlues or ll possile sis input vetors re on sinle ritrry irle C wit te oriin s its enter E point on irle C n e represented s were is vetor pssin tru te oriin nd perpendiulr to irle C In ddition, is te quntum - deree rottion opertion round te vetor nd ˆ0 n e set to ny point on te irle C Fiure 9 Su untion n e represented wit QDD were te terminl node is ˆ0 nd te weits o edes nd te root node re in te orm o Notie tt te quntum iruit syntesized y usin te q-tor loritm s te property tt or te sis input vetors, te vlues o ll internl nd output sinls will lie on some irle, C Te inl tored orm resultin rom q-pply s te orm were is te input vetor Sine untions i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso e represented s: were i vlues re rerded s rel vlues 0 nd ˆ Let s deine te rel vlued untion s Consequently, untion n e represented s Te struture o te syntesized iruit n e represented s Fiure 0 Note tt G is te inverse untion o G wit rerd to X Y ˆ Z C Fiure 9 Blo spere representtion G G Fiure 0 Quntum iruit perormin \ G G ˆ0 \ γ Fiure : sdsd 8 loritm n esily e enerlized to e pplile to quntum untions wose output vlues or ll possile sis input vetors re on sinle ritrry irle C wit te oriin s its enter E point on irle C n e represented s were is vetor pssin tru te oriin nd perpendiulr to irle C In ddition, is te quntum - deree rottion opertion round te vetor nd ˆ0 n e set to ny point on te irle C Fiure 9 Su untion n e represented wit QDD were te terminl node is ˆ0 nd te weits o edes nd te root node re in te orm o Notie tt te quntum iruit syntesized y usin te q-tor loritm s te property tt or te sis input vetors, te vlues o ll internl nd output sinls will lie on some irle, C Te inl tored orm resultin rom q-pply s te orm were is te input vetor Sine untions i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso e represented s: were i vlues re rerded s rel vlues 0 nd ˆ Let s deine te rel vlued untion s Consequently, untion n e represented s Te struture o te syntesized iruit n e represented s Fiure 0 Note tt G is te inverse untion o G wit rerd to X Y ˆ Z C Fiure 9 Blo spere representtion G G Fiure 0 Quntum iruit perormin 8 usin te q-tor loritm s te property tt or te sis input vetors, te vlues o ll internl nd output sinls will lie on some irle, C Te inl tored orm resultin rom q-pply s te orm were is te input vetor Sine untions i only te vlues nd ˆ or sis input vetors, it n e seen tt n lso e represented s: were i vlues re rerded s rel vlues 0 nd ˆ Let s deine te rel vlued untion s Consequently, untion n e represented s Te struture o te syntesized iruit n e represented s Fiure 0 Note tt G is te inverse untion o G wit rerd to Fiure 0 Quntum iruit perormin G G' ' Fiure : Blo spere representtion Quntum iruit perormin = [γ Te inl tored orm resultin rom q-pply s te orm = γ [ γ were is te input vetor Sine untions i only te vlues ˆ0 nd ˆ or sis input vetor e seen tt n lso e represented s = [ γ + γ + + γ ] i vlues re rerded s rel vlues ˆ0 0 nd ˆ Lets deine te rel vlued γ s γ = γ + γ + + γ Consequently, untion n e repres = [γ] ˆ0 Te struture o te syntesized iruit n e represented s Fiure tt G is te inverse untion o G wit rerd to Te rit portion o te iruit is need i it is required tt quits ssoited wit input lines mintin teir initil vlue To l roles o G nd G, it will e eneiil to ompre tis iruit wit te tree-input multiplee = s + s syntesized y te q-tor loritm in Fiure 3 I insted o ˆ0, noter q vlue q is used in tis iruit s te initil vlue or te input, ten te resultin iruit will im te ollowin untion = [γ] q Now we enerlize te syntesis ppro or syntesizin quntum untions tt or iven vetors enerte enerl quntum vlue = [ 0 ] T Beuse 0 + we my rewrite s = e iδ [ e iγ/ os θ ie iγ/ sin θ ] T Hene, epressed s = e iδ zγ θˆ0 were z is te rottion opertor round te deined s zγ = [ e iγ/ 0 0 e iγ/ ] We n inore te term e iδ sine it s no oservle nd tereore we n eetively write = zγ θˆ0 see Fiure 4 Tis on s s ˆ0 / / / / Fiure 3: Quntum tree-input multipleer 0 Fiure : Blo spere representtion Quntum iruit perormin = [γ] ˆ0 \ G G G G ˆ0 \ θ zγ Fiure : sd Te inl tored orm resultin rom q-pply s te orm = γ [ γ [ [ γˆ0 ] ] ] were is te input vetor Sine untions i only te vlues ˆ0 nd ˆ or sis input vetors, it n e seen tt n lso e represented s = [γ + γ + + γ] ˆ0 were i vlues re rerded s rel vlues ˆ0 0 nd ˆ Lets deine te rel vlued untion γ s γ = γ + γ + + γ Consequently, untion n e represented s = [γ] ˆ0 Te struture o te syntesized iruit n e represented s Fiure Note tt G is te inverse untion o G wit rerd to Te rit portion o te iruit is needed only i it is required tt quits ssoited wit input lines mintin teir initil vlue To lriy te roles o G nd G, it will e eneiil to ompre tis iruit wit te tree-input multipleer iruit = s + s syntesized y te q-tor loritm in Fiure 3 I insted o ˆ0, noter quntum vlue q is used in tis iruit s te initil vlue or te input, ten te resultin iruit will implement te ollowin untion = [γ] q Now we enerlize te syntesis ppro or syntesizin quntum untions tt or iven sis input vetors enerte enerl quntum vlue = [ 0 ] T Beuse 0 + =, we my rewrite s = e iδ [ e iγ/ os θ ie iγ/ sin θ ] T Hene, n e epressed s = e iδ zγθˆ0 were z is te rottion opertor round te Z is deined s zγ = [ e iγ/ 0 0 e iγ/ ] We n inore te term e iδ sine it s no oservle eets [] nd tereore we n eetively write = zγθˆ0 see Fiure 4 Tis onept n s s ˆ0 / / / / Fiure 3: Quntum tree-input multipleer 0 Fi 7 Quntum iruit perormin = γˆ0 Quntum 3-input multipleer Dsed oes represent G, G, nd γ in Only rottion nles re reported or θ tes Quntum iruit perormin = zγ θˆ0 iruit n e represented s Fi 7 In tis iure, G is iruit tt onstruts γ nd G is te inverse o G Note tt G sould e used only i one wnts to eep input lines unned To lriy te roles o G nd G, see te 3-input multipleer iruit = s + s syntesized y te tor loritm in Fi 7 I insted o ˆ0, noter quntum vlue q is used in tis iruit s te initil vlue or te input, te resultin iruit implements = [γ] q Te onstnt nill reister in Fi 7 my not e neessry in some se For emple, te ontrolled rottion wit ontrol quit nd tret ˆ0 enertes s te seond output nd te use o te ontrolled rottion in tis se is unneessry ie, ˆ0 = Setion 6 sows severl emples Now onsider iven untion tt or iven sis input vetors enertes enerl vlue = [ 0 ] T Sine 0 + =, we my rewrite s: = e iδ [ e iγ/ os θ ie iγ/ sin θ ] T Hene, n e epressed s = e iδ z γ θˆ0 were z is te rottion opertor round te z is We n inore te lol pse e iδ sine it s no oservle eets [] Tereore, one n eetively write = z γ θˆ0 Note tt z γ θˆ0 results rom θ rottion o ˆ0 round te is ollowed y γ rottion round te z is in te Blo spere Te quntum iruit or = z γ θˆ0 n e syntesized s: Syntesize = θˆ0 y usin te tor loritm Syntesize = z γˆ0 y usin te tor loritm Csde te resultin iruits s depited in Fi 7 In tis iure, G nd G re or nd, respetively Aordinly, G nd G re te inverse iruits o G nd G 6 esults Multiple-ontrol Tooli te Consider 4-input Tooli te in Fi 8 nd te DD o te tret output in Fi 8 wit vrile orderin < < < d Comprin te DD o s wit te enerl DD struture in Fi 6 revels tt vrile orresponds to v Additionlly, r-de=, α = 0 nd α = wi result in γ = / Teorem 7 7 One my set α = nd α = 0 Tis omintion enertes dierent iruit wit te sme untionlity